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A077177
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Number of primitive Pythagorean triangles with perimeter equal to A002110(n), the product of the first n primes.
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0
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0, 0, 1, 0, 1, 2, 3, 5, 8, 17, 34, 59, 111, 213, 396, 746, 1413, 2690, 5147, 9826, 18885, 36269, 69952, 134949, 260743, 504636, 978311, 1899832, 3692980, 7190329, 13994206, 27279898, 53195986
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OFFSET
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1,6
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COMMENTS
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A Pythagorean triangle is a right triangle whose edge lengths are all integers; such a triangle is 'primitive' if the lengths are relatively prime.
Equivalently, number of divisors of s=A070826(n) in the range (sqrt(s), sqrt(2s)). More generally, for any positive integer s, the number of primitive Pythagorean triangles with perimeter 2's equals the number of odd unitary divisors of s in the range (sqrt(s), sqrt(2s)). (A divisor d of n is 'unitary' if gcd(d,n/d)=1.)
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REFERENCES
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A. S. Anema, "Pythagorean Triangles with Equal Perimeters", Scripta Mathematica, vol. 15 (1949) p. 89.
Albert H. Beiler, "Recreations in the Theory of Numbers", chapter XIV, "The Eternal Triangle", pp. 131, 132.
F. L. Miksa, "Pythagorean Triangles with Equal Perimeters", Mathematics, vol. 24 (1950), pg 52.
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LINKS
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Table of n, a(n) for n=1..33.
Randall L. Rathbun, Equal Perimeter primitive right triangles
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FORMULA
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a(n) = A070109(A002110(n)) = A078926(A070826(n)).
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EXAMPLE
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a(5) = 1 since there is exactly one primitive Pythagorean triangle with perimeter 2*3*5*7*11; its edge lengths are (132, 1085, 1093). a(7) = 3; the 3 triangles have edge lengths (70941, 214060, 225509), (96460, 195789, 218261) and (142428, 156485, 211597).
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MATHEMATICA
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a[n_] := Length[Select[Divisors[s=Times@@Prime/@Range[2, n]], s<#^2<2s&]]
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PROG
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(PARI) semi_peri(p)= {local(q, r, ct, tot); ct=0; tot=0; pt=0; fordiv(p, q, r=p/q-q; if(r<=q&&r>0, print(q, ", ", r, " [", gcd(q, r), "] "); if(gcd(q, r)==1, ct=ct+1; if(q*r%2==0, pt=pt+1; ); ); tot=tot+1); ); print("semi-perimeter:"p, " Total sets:", tot, " Co-prime:", ct, " Primitive:", pt); } /* Lists all pairs q, r such that the triangle with edge lengths (q^2-r^2, 2qr, q^2+r^2) has semi-perimeter p. */
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CROSSREFS
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Cf. A002110, A070109, A070826, A078926.
Sequence in context: A122630 A108054 A123612 * A145793 A113879 A205303
Adjacent sequences: A077174 A077175 A077176 * A077178 A077179 A077180
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KEYWORD
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more,nonn
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AUTHOR
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Kermit Rose and Randall L. Rathbun, Nov 29 2002
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EXTENSIONS
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Edited by Dean Hickerson (dean.hickerson(AT)yahoo.com), Dec 18 2002
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STATUS
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approved
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